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International Contests
Austrian-Polish
1984 Austrian-Polish Competition
3
An inequality with n-variables
An inequality with n-variables
Source: Austrain Polish 1984
December 7, 2013
inequalities
inequalities unsolved
Problem Statement
Show that for
n
>
1
n>1
n
>
1
and any positive real numbers
k
,
x
1
,
x
2
,
.
.
.
,
x
n
k,x_{1},x_{2},...,x_{n}
k
,
x
1
,
x
2
,
...
,
x
n
then
f
(
x
1
−
x
2
)
x
1
+
x
2
+
f
(
x
2
−
x
3
)
x
2
+
x
3
+
.
.
.
+
f
(
x
n
−
x
1
)
x
n
+
x
1
≥
n
2
2
(
x
1
+
x
2
+
.
.
.
+
x
n
)
\frac{f(x_{1}-x_{2})}{x_{1}+x_{2}}+\frac{f(x_{2}-x_{3})}{x_{2}+x_{3}}+...+\frac{f(x_{n}-x_{1})}{x_{n}+x_{1}}\geq \frac{n^2}{2(x_{1}+x_{2}+...+x_{n})}
x
1
+
x
2
f
(
x
1
−
x
2
)
+
x
2
+
x
3
f
(
x
2
−
x
3
)
+
...
+
x
n
+
x
1
f
(
x
n
−
x
1
)
≥
2
(
x
1
+
x
2
+
...
+
x
n
)
n
2
Where
f
(
x
)
=
k
x
f(x)=k^x
f
(
x
)
=
k
x
. When does equality hold.
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